Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
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Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) ...
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441 views

Discrete Time Fourier Transform example: $x = [1 \; 2 \; 3 \; 4]^T \; \rightarrow \; X=?$

How do I find the Discrete Fourier Transform of the sequence below? $$ x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ Show all steps.
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84 views

Why aren't these two question equal?

Firstly I doubt whether the 12 is right in Q1.If it is right,please give a proof. Secondly why (1) is not equal to (2) in Q2?
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73 views

I can't understand the last second step.

It is known that $f(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{inx}$, with $c_{n}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\:dx$, for $n\in\mathbb{Z}$. To prove: ...
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160 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
2
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1answer
65 views

Fourier Transform is onto $L^2(\mathbb{R})$

In Introduction to Fourier Analysis on Euclidean Spaces, the authors explore the $L^2$ extension of the Fourier transform and argue that it is onto $L^2(\mathbb{R})$ but I can't follow their ...
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118 views

Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
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118 views

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
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38 views

nasty exponentials

While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like $$\large \rm ...
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92 views

Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$

Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ (a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
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55 views

Show that Fourier coefficients approach zero uniformly

Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that $$ a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
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193 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
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Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
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1answer
73 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
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Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
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Fast Fourier Transform of 2D axisymmetric geometry

I need to compute the FFT of a signal in 2D axisymmetric geometry. The signal consists of a snapshot in time of a laser beam for which I have values in z (direction of propagation) and r (from 0 to ...
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Solving an integral equation using the Fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
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93 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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1answer
76 views

Fourier transforms - don't understand this concept!!! Please help me on this

I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help ...
0
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1answer
71 views

Fourier transform parameter

I'm currently working on the problem: If $\hat{f}(k)$ is the complex Fourier Transform of the function $f(x)$ and $a$ is a real constant with $a>0$, show that the complex Fourier Transform of ...
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Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
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Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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1answer
440 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
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1answer
130 views

Unclear on relationship between different dimensionalities of Fourier transform

This is probably a silly question, but it's one that's directly relevant to a project of mine and I figured this was the place to go. I have some objects that contain a 1d and a 2d array of double ...
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Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?

If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
14
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1answer
532 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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1answer
51 views

A function whose derivatives always have a convergent fourier series

I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
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275 views

Fourier transform of $(1-\cos(tx))/x^2$

I am trying to compute the Fourier transform of $f(x) = \frac{1-\cos(tx)}{x^2}$, $(t > 0)$ directly. I tried contour integration, and could not seem to get it to work. So, I am wondering if it can ...
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716 views

How to reperesent $\sin^{4}(x)$ byFourier series? [closed]

how to represent $\sin^{4}(x)$ by Fourier series? Obviously,$\sin^{4}(x)$ is an even function, so $b_n=0$. How can i get $a_n$ ?
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132 views

How to solve this equation by Fourier series?

$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$ Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
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1answer
63 views

I need to find the Fourier transform of the following function

I need to find the Fourier transform of $e^{-2 \pi |x|}$. Normally, I can do something like this, but the absolute value is kind of confusing me.
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Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
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Upper bound on truncation error of a fourier series approximation of a pdf?

Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation: $$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$ ...
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53 views

Flipping and convolution theorem

Could you provide the mathmatical proof that multiplication in Fourier domain is only convolution , when the flipping (of one of the signals/functions) occurs. So, that multiplication is not ...
2
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1answer
494 views

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ ...
3
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1answer
81 views

Understanding fourier notation $F(\partial_x)$

Can somebody please help me understand some of the notion in the equations below, taken from a published paper on image de-blurring. I have an energy $E(H)$ defined over an image $H$, a point-spread ...
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What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
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195 views

Pointwise convergence of double Fourier series

I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions. Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite ...
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2answers
73 views

Computing the Fourier transform of a certain function

Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ be the function defined by $f(x)=(1+|x|^2)^{-1}$. Problem: How can I find $\hat{f}$ by direct computation? Remark: This is exercise 8.5 in Rudin's ...
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1answer
2k views

$b_n$ Fourier series $f(x)=x^3$

I'm having trouble to solve the $b_n$ coefficient of Fourier series of $x^3$ function on the $-\pi<x<\pi$. The result is almost right, but the powers of $n$ are one less degree I mean The ...
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236 views

Exponential Form of Fourier Series

Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...
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Arnold's Trivium problem 51

Calculate $$ f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$ As far as I know, this is not a function but rather the Fourier transform in tempered distributions. 1) What is ...
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1answer
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Solving Wave equation using Fourier Transforms

My approach is the following: we take Fourier transform with respect to $x$, where $k$ is the variable the resulting fourier transform is in. $\hat{u}_{tt} + k^2 \hat{u} = 0$ Solving this gives me ...
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1answer
435 views

Schonhage–Strassen algorithm

After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
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1answer
96 views

Fourier transform in $L^2$

I have a function $\phi\in L^2(\mathbb{R}^3)$ and I know that its Fourier transform satisfies the following equation: $$(p^2-A)\hat{\phi}(p)=Q\frac{A+\lambda}{p^2+\lambda}$$ where $Q$ is a constant, ...
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212 views

DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?

I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation: Say, I have a function vector with ...
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1answer
516 views

Confused as to how to prove the basis of dft is orthonormal

I have been stuck for hours trying to prove that the basis of discrete fourier transform is orthonormal can anyone point me in the direction of how to do so
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102 views

Inversion formula for Schwartz-space $\mathcal{S}$.

Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...